In this talk we will survey on recent results related to the
existence and regularity of solutions to nonlocal differential equations
of different type. The topic is intrinsically connected to probability
and Markov processes with jumps and was revitalized a few years ago by
Luis Caffarelly and his many collaborators. In many cases, the
integro-differential operator is modeled by the fractional Laplacian.
We will concentrate in three examples that show similarities and
differences with respect to standard local PDE: the regularity of
solutions for the fractional porous medium flow; well posedness for
positive solutions to the heat equation with nonlocal diffusion, and an
eigenvalue nonlocal mixed problem in bounded domains. The results
coming from these examples can be found in the following publications:
[1] L. Caffarelli; F. Soria; JL Vazquez; "Regularity of solutions of
the fractional porous medium flow". J. Eur. Math. Soc. (JEMS) 15 (2013),
no. 5
[2] B. Barrios; I. Peral; F. Soria; E. Valdinoci; "A Widder's type
theorem for the heat equation with nonlocal diffusion". Arch. Ration.
Mech. Anal. 213 (2014)
[3] T. Leonori; M. Medina; I. Peral; A. Primo; F.Soria; "Principal
eigenvalue of mixed problem for the fractional Laplacian: Moving the
boundary conditions". J. Differential Equations (2018)
